Levels of Measurement and Cardinal Utility

A few weeks ago, I was having a chat with Todd and some others in the office and it was in the
conversational mix that cardinal utility had the property of preserving “intervals.” It was
occasionally also mentioned that such utility representations were closed under
“linear transformations.” I was confused by the discussion and at first I didn’t know why.
On my walk home that day, I remembered I had heard those sorts of claims before. I typically
think of a linear transformations as any mapping from a vector space to a vector
space , both over a field , with the following properties of linearity:

if is in the field then for , ;

if , then .

For example, the equation is a linear transformation from the vector space of the
set of reals back into itself. So, . When
we speak of the algebra on in one dimension, is the underlying set for
the vector space as well as the field.

Note that has the first property; suppose for example that and . Then

It also has the second property; for example, let and ; then

But clearly, this linear transformation does not preserve intervals:

I didn’t think I could be wrong about my understanding of the conventional use of the term
“linear.” I thought maybe what people mean instead of “linear” in this context is that cardinal
utility was closed under the class affine transformations. That is, the class containing
all transformations of the following form (where is a scalar value in the field of
and ):

The class of affine transformations is a proper superclass of the class of linear transformations.
So the class of affine transformations does not always preserve intervals since is an affine
transformation with set to the zero vector. It is easy to see that the class of affine
transformations does not preserve ratios, for example let . If we let
and , we have that:

To make matters somewhat worse, it appears that the form of is quite correctly called
the form of a linear equation. However, some linear equations are not linear
transformations because they fail to have the property of additivity. To wit:

Perhaps this subtle confusion between linear equations and linear transformations leads to a
terminological confusion which makes people say that cardinal utility is closed under positive
linear transformations, as in the final section of this
Wikipedia article on cardinal utility (where cardinal utility is
construed as von Neummann-Morgenstern utility).

Both our example transformations, and preserve the order of the reals.
That is, for , we have that

and

Note that this property, positive monotonicity, is a consequence of the
fact that and . One compelling feature of positive
monotone transformations is that they preserve an existing order. However if a kind of utility
only preserves an underlying order, it is typically regarded as ordinal utility in
contrast to cardinal utility.

The class of translations does preserve intervals. These are the class of transformations of the
following form (for ):

The effect of this transformation is to move a vector space in some direction (to the “left” or
“right” by some value . For example let . Then if and ,

But if a utility measurement is only closed under translations, then that closure
condition is tantamount to the view that there is something absolute about scale. It is not
disputed that von Neumann-Morgenstern utility is closed under more than simply translations.

I think all the while the problem is that it is simply cumbersome in conversation to say which
property is preserved by the affine transformations. It is neither ratios nor intervals but
rather ratios of intervals. To see this, consider the following proposition.

Claim. Let and let be affine.
Then

Proof.
Continuing we have that

as desired.∎

The following table summarizes what I think the correct picture should be:

InvarianceClass

PropertyPreserved

ExampleTransformation

MeasurementExample

Ordinal Scale

PositiveMonotone

Order

Ordinal Utility

Ratio Scale

PositiveLinear

Order, Ratios, Interval Ratios,Zero Point

Mass, Length

Difference Scale

Translation

Order, Intervals, Interval Ratios

Interval Scale

PositiveAffine

Order, Interval Ratios

Celsius,vNM Utility

Table 1. Invariance Classes of Common Scales

If all of the above is correct then it should be appear that it is technically incorrect to ever
say that cardinal utility has the invariance property of preserving intervals even though we may
call such utilities, perplexingly, an interval scale. Furthermore, while a cardinal
utility like a vNM utility is closed under positive linear transformations, its invariance
class is actually more properly understood as the class of positive affine transformations.